Newton’s method

Definition

Newton's Method is an iterative numerical technique used to approximate the roots of a real-valued function. It involves using the derivative of the function to iteratively converge on a root starting from an initial guess.

5 Must Know Facts For Your Next Test

1. Newton’s Method requires an initial guess, and its accuracy depends heavily on this choice.
2. The formula for Newton's Method is $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$.
3. Convergence is typically quadratic if the method converges, meaning errors decrease exponentially with each iteration.
4. If the derivative $f'(x)$ is zero or close to zero, Newton’s Method may fail or provide inaccurate results.
5. Newton's Method can diverge if the initial guess is not sufficiently close to the actual root.

Review Questions

• What formula represents Newton's Method?
• Why might Newton’s Method fail to converge?
• How does quadratic convergence benefit Newton's Method?

Related terms

Derivative: A measure of how a function changes as its input changes; it represents the slope of the function at any point.

Root-finding Algorithm: A computational algorithm used to find solutions where a given function equals zero.

Quadratic Convergence: A property of an iterative method where errors decrease proportionally to the square of previous errors in successive iterations.